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### WS21_Class%20notes%20for%20continuous%20random%20variable_students

Course: MATH 115, Fall 2009
School: Arizona
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Word Count: 667

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Class WS note on continuous random variable A random variable whose possible values form an interval is called a continuous random variable. For c.d.f. cumulative distribution function, FX ( x ) = P ( X x ) any continuous random variable Since the random variable is continuous the growth [0,1] is a continuous curve. There are no jumps. For a continuous random variable, the probability at any particular value is...

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Class WS note on continuous random variable A random variable whose possible values form an interval is called a continuous random variable. For c.d.f. cumulative distribution function, FX ( x ) = P ( X x ) any continuous random variable Since the random variable is continuous the growth [0,1] is a continuous curve. There are no jumps. For a continuous random variable, the probability at any particular value is zero. P ( X = a ) = 0 for all a. We measure probability on intervals P ( a x b) = FX ( b ) FX ( a ) 091 P (a < x b) = FX ( b ) FX ( a ) P (a x < b) = FX ( b ) FX ( a ) P (a < x < b) = FX ( b ) FX ( a ) Since the probability at any particular value is zero the and < makes no difference to the probabilities f X ( x ) is the probability density function. This is just the height of the function. The area under the curve equals 1. The height is not the probability at a point. We only can measure probability in intervals. The area under the curve between any two values is the probability that the event has occurred in that interval. Example1: This is the f X ( x ) (p.d.f. ) a) Show the P (1.5 < X 4) on the graph b) What is P ( X = 1) c) Estimate P (1.5 < X 4) Example 2: The following is a pdf: f(x) and its cdf: F(x) for a continuous random variable function a. Make an interval (a,b) on both graphs (on the horizontal) b. Show the P(a x b) on both graphs f(x) F(x) x x Inclass Continuous Random variable pg. 2 Uniform continuous random variable Example 3 The security guard will pass by my office between 12 -2pm. Each moment of time is equally likely. What is the probability that he will pass by my office between 12:15 and 1:12pm today? Need to find the amount of hours past noon: 12:15 is 0.25 of an hour and 1:12 is 1.2 hours Graph the p.d.f Write the function f X ( x) Graph the c.d.f. Write the function FX ( x ) Evaluate the following b. f X (1.2 = ) a. f X (.25 ) = c. P ( X = .25) = d. P ( X = 1.2) e. = f. FX (2) = g. FX (.25) = h. FX (1.2) = Now we will answer the question: What is the probability that he will pass by my office between 12:15 and 1:12pm today? Find P (.25 X 1.2) There are two ways to calculate this Inclass Continuous Random variable pg. 3 Exponential random variable Example 4 Suppose that X is an exponential random variable, which has the following p.d.f. and c.d.f. 0 f X ( x ) = 1 7x 7 e a) Sketch x<0 x0 x<0 0 FX ( x ) = x 1 e 7 x 0 Sketch f X ( x ) graph. FX ( x ) graph. Use your graphing calculator Use your graphing calculator b) Calculate (5 decimal approximation) c) Calculate (5 decimal approximation) Show the exact value of each probability and then give the 4 decimal place approximate is appropriate d) Sketch show and calculate this value e) What is the probability that f) Calculate Only one method X=3 g) What is the probability that the random variable is greater than 4 h) What is the mean of this distribution? i) Is f X ( 4 ) = P ( X = 4) ? Explain Inclass Continuous Random variable pg. 4 Expected V...

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